Transcript Augmented Designs

Types of Checks in Variety Trials
 One could be a long term check that is unchanged from
year to year
– serves to monitor experimental conditions from year to year
– is a baseline against which to measure progress
 Other checks may be included for different purposes
– a “local” variety would be good if comparing diverse locations
– might want a susceptible to get a baseline for disease
expression
– a new variety could serve as the “best” current standard
Replication of Checks
 Because all new entries are compared to the
same checks, the checks should be
replicated at a higher rate than any of the
new entries
– number of replications of a check should be the
square root of the number of new entries in the
trial
– so if you had 100 new entries, you would need 10
replications of the check for each replication of
the new entries.
LSI = t a
(rc+ r )MSE / rcr
rc=replications of checks
r =replications of new entries
Early Stage Yield Trials
 Seed is precious - in early stages, usually not enough to
replicate
 Could plant small plots (often single rows) and at regular
intervals plant a check
– consider how many adjacent plots are likely to be grown under
uniform conditions, given the soil heterogeneity, and the
sensitivity of the crop and response variables to environmental
factors; plant a check at appropriate intervals
– could make subjective comparisons of new entries with nearest
check
– alternatively, get an estimate of experimental error from the
variation among the checks. Then compute an LSI to compare
the yields of the new lines to the checks
LSI = t a
(rc+ 1)MSE / rc
Early Stage Yield Trials
 But there are disadvantages
– the checks are often systematically placed, so
estimate of experimental error may not be
valid
– no provision is made to adjust yields for
differences in soil, etc.
Augmented Designs – An alternative
 Introduced by Federer (1956)
 Controls (check varieties) are replicated in a
standard experimental design
 New treatments (genotypes) are not replicated,
or have fewer replicates than the checks – they
augment the standard design
Augmented Designs - Advantages
 Provide an estimate of standard error that can be
used for comparisons
– Among the new genotypes
– Between new genotypes and check varieties
 Observations on new genotypes can be adjusted
for field heterogeneity (blocking)
 Unreplicated designs can make good use of scarce
resources
 Fewer check plots are required than for designs
with systematic repetition of a single check
 Flexible – blocks can be of unequal size
Some Disadvantages
 Considerable resources are spent on production
and processing of control plots
 Relatively few degrees of freedom for experimental
error, which reduces the power to detect
differences among treatments
 Unreplicated experiments are inherently imprecise,
no matter how sophisticated the design
Applications of Augmented Designs
 Early stages in a breeding program
– May be insufficient seed for replication
– Using a single replication permits more genotypes to
be screened
 Participatory plant breeding
– Farmers may prefer to grow a single replication when
there are many genotypes to evaluate
 Farming Systems Research
– Want to evaluate promising genotypes (or other
technologies) in as many environments as possible
Augmented Design in an RBD
 Area is divided into blocks
– these are incomplete blocks because they contain only a subset of
the entries
 Two or more check varieties are assigned at random to
plots within the blocks
– same check varieties appear in each block
– little is lost if you want to place one check systematically - a block
marker
 Most efficient when block size is constant
 Checks are replicated, but new entries are not
So how many blocks?
 Need to have at least 10 degrees of freedom for error in
the ANOVA of checks
 df for error = (r-1)(c-1)
– c=number of different checks per block
– r=number of blocks=number of replicates of a check
 Minimum blocks would be r > [(10)/(c-1)] + 1
 For example, with 4 checks
[(10)/(4-1)]+1=(10/3)+1=3.33+1=4.33 ~ 5
you would need 5 blocks
 Each block has at least c+1 plots
Analysis
 Experimental error is estimated by treating the
checks as if they were treatments in a RBD
 MSE is then used to construct standard errors
for comparisons
 Adjustments for block differences
– based on difference between block check means and over-all
check mean*
– Recall Yij =
_  + Bi + Tj + eij
– ai = Xi - X
– therefore Siai = 0
*this calculation assumes that blocks are fixed effects
(we will use this simplification to illustrate the concept)
Steps in the Analysis
 Construct a two way table of check variety x block
means
 Compute the grand mean and the mean of the
checks in each block
 Compute the block adjustment as
a i = Xi - X
 Adjust yields of new selections as
^
Y ij = Y ij - a i
 Complete a standard ANOVA (RBD) using check
yields
ANOVA
Source
Total
Blocks
Checks
Error
df
rc-1
r-1
c-1
(r-1)(c-1)
SS
MS
(
SSTot =  i  j Yij - Y
( )
SSC = r  ( Y - Y )
SSR = t  i Y i - Y
j
)
2
2
2
j
SSE = SSTot - SSR - SSC MSE=SSE/dfE
Standard Errors
 Difference between two check varieties
s c = 2MSE / r
 Difference between adjusted means of two selections in
the same block
s d = 2MSE
 Difference between adjusted means of two selections in
different blocks
s v = 2(c + 1)MSE / c
 Difference between adjusted selection and check mean
s vc = ((r + 1)(c + 1)MSE ) / rc
c=number of different checks per block
r=number of blocks=number of replicates of a check
Numerical Example
 Testing 30 new selections using 3 checks
 Number of blocks:
– ((10)/(c-1))+1 = (10/2)+1 = 6
 Number of selections per block:
– 30/6 = 5
– Randomly assign selections to blocks
 Total number of plots
– (5+3)*6=48
Field Layout
I
II
III
IV
V
VI
C1
C1
C1
C1
C1
C1
V14
C2
V18
V9
V2
V29
V26
V4
V27
V6
V21
V7
C2
V15
C2
C2
C3
C2
V17
V30
V25
C3
C2
V1
C3
V3
V28
V20
V10
C3
V22
C3
V5
V11
V8
V12
V13
V24
C3
V23
V16
V19
C1 is placed systematically first in each block as a “marker”
RBD analysis of check means
Source
df
SS
Total
17
7,899,564
Blocks
5
6,986,486
Checks
2
20,051
10
911,027
Error
MS
estimate of
experimental
error to be used
in LSI
computation
91,103
Yields, Totals, and Means of Checks
Variety
I
II
III
IV
V
VI
Mean
C1
2972
3122
2260
3348
1315
3538
2759
C2
2592
3023
2918
2940
1398
3483
2726
C3
2608
2477
3107
2850
1625
3400
2678
Mean
2724
2874
2762
3046
1446
3474
2721
3
153
41
325
- 1275
753
Adjust
se difference between 2 adj means of selections in different blocks =
(2(c + 1)MSE ) / c = (2(3 + 1)(91103)) / 3 = 493
se difference between adjusted selection mean and check=
((r + 1)(c + 1)MSE ) / rc = (6 + 1)(3 + 1)(91103) / (6 * 3) = 376
t value has (r-1)(c-1) = 10 df
A Comparison Statistic
 Because we are looking for those that exceed the
check, we compute LSI
– 1-tailed t with 10 df at α=5% = 1.812
– LSI = (1.812) ((6+1)(3+1)(91103)/(6*3) =
1.812*376=681
 Any adjusted selection greater than
– 2759+681=3440 significantly outyields C1
– 2726+681=3407 significantly outyields C2
– 2678+681=3359 significantly outyields C3
Selection
11
21
3
19
4
26
27
30
25
16
Stork
Adj Yield
Selection
Adj Yield
3055
2963
2902
2890
2865
2852
2816
2802
2784
2770
2759
Cimmaron
22
Waha
24
17
10
18
8
7
23
14
2726
2702
2678
2630
2569
2568
2562
2528
2512
2445
2402
Selection
Adj Yield
13
20
2
15
1
29
5
9
28
6
12
2388
2345
2330
2324
2260
2162
2024
1943
1862
1823
1632
The standard error of the difference between adjusted selection yield
and a check mean
s =
vc
((r + 1)(c + 1)MSE ) / rc = (6 + 1)(3 + 1)(91103) / (6 * 3) = 376
Compute the LSI using a 1-tailed t and 10 degrees of freedom (MSE)
LSI = t a s vc = (1.812)(376) = 681
Stork
Cimmaron
Waha
2759+681
2726+681
2678+681
3440
3407
3359
Interpretation
 Although the adjusted yield of 10 of the new
selections was greater than the yield of the
highest check, C1, none of the yields was
significantly higher than any of the check means
Variations in Augmented Designs
 New treatments may be considered to be fixed or
random effects
– best to use mixed model procedures for analysis
 Can adjust for two sources of heterogeneity
using rows and columns
 Modified designs use systematic placement of
controls
 Factorials and split-plots can be used
 Partially replicated (p-rep) augmented designs
use entries rather than checks to estimate error
and make adjustments for field effects